Good try! Your interpretation isn’t quite right. Given a fixed atlas, you can construct a maximal atlas “by adding to it every compatible chart”. In this case, $\{U_\alpha, \phi_\alpha\}$ and $\{V_\beta, \psi_\beta\}$ are atlases of two different manifolds, $M$ and $N$, respectively.
This is a very simple “definition chase”. Let’s prove (a) implies (b). Since $M$ and $N$ are manifolds, they have fixed atlases $\{U, \phi\}$ and $\{V, \psi\}$. If (a) holds, then for every point $p$, there exist charts (of the fixed atlases) containing $p$ and $F(p)$ such that the stated triple composition is smooth. Trivially, setting $\{U, \phi\}$ = $\{U_\alpha, \phi_\alpha\}$, and $\{V, \psi\}$ = $\{V_\beta, \psi_\beta\}$, we get that (b) holds. Conversely, to show (b) implies (a), you do the definition chase the other way around, it should be easy.
The requirement that $F$ is continuous in (b) is equivalent to saying that the pre-image of every open set is itself open. So trivially, the requirement that $F$ is continuous in (b) implies that $F^{-1}(V)$ is open in $M$ and hence $F^{-1}(V)\cap U$ is open in $M$. Conversely, we need to check $F^{-1}(V)$ is open in $M$ for each $V$ in the atlas of $N$, using the condition in (a). Starting from all the conditions in (a), instead of setting $\{V_\beta, \psi_\beta\}\equiv\{V, \psi\}$ , we set $\{V_\beta, \psi_\beta\}$ to be the maximal atlas containing $\{V, \psi\}$. Then, it is a lemma that the maximal atlas contains a basis for the topology of $N$ (which, in broad strokes, follows from the fact that the maximal atlas contains the so-called “coordinate balls” in Lee’s terminology). It is a fact that it suffices to check that the pre-image of every open set of a basis for the topology of $N$ is open, in order to establish that $F$ is a continuous function. So we can check $F^{-1}(V)$ is open in $M$ for every $V$ in the maximal atlas. But we can see that $F^{-1}(V)$=$F^{-1}(V)\cap M$= $F^{-1}(V)\cap (\cup_i U_i)$ = $\cup_i (F^{-1}(V)\cap U_i)$ which is a union of open sets by the condition in (a) that we are assuming. Hence, $F$ is continuous as (b) requires.
In 2.6, you don’t need to endow each $U$ with a submanifold structure. It can be a single-chart manifold and you can check smoothness directly. In terms of the interpretation, it is telling you that smoothness is a local condition. An easier example to think about is a function from the reals to the reals which is globally $C^1$, by definition this means that it the derivative exists and is continuous at every point in the reals. It is trivial to see that globally $C^1$ implies locally $C^1$, and vice versa.
